Abstract

Four measurement principles based on the theory of fluctuation in the regular transmittance through a dispersion of large cylindrical particles are presented. The principles concern the measurement of particle concentration, particle length classes, particle length, and particle width. In previous publications, the measurement principles were briefly presented. A more complete description, experimental demonstrations, and a number of considerations needed to understand the region of validity of the different principles, as well as their uncertainty, are presented. A recently published simulation model [Appl. Opt. 47, 993 (2008)] is used. The basic ideas of the measurement principles within their respective regions of validity are supported.

© 2008 Optical Society of America

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  1. C. E. Hubley, A. A. Robertson, and S. G. Mason, “Flocculation in suspensions of large particles,” Can. J. Res. 28, 770-787(1950).
  2. A. A. Robertson and S. G. Mason, “Flocculation in flowing pulp suspensions,” Pulp Pap. Mag. Can. 55, 263-269 (1954).
  3. T. Pettersson, G. Fladda, and L. Eriksson, “Förfarande för koncentrationsbestämning,” Swedish patent 75 135 24-4 (29 September 1977).
  4. T. Pettersson, G. Fladda, and L. Eriksson, “Method for determination of concentration,” U.S. patent 4,110,044 (29 August 1978).
  5. T. Pettersson and H. Karlsson, “Förfarande för att bestämma medelpartikelradie och/eller medelpartikellängd,” Swedish patent 81 058 02-6 (21 April 1988).
  6. T. Pettersson and H. Karlsson, “Method for determining the average radius and/or the average length of particles carried by a flowing medium,” U.S. patent 4,529,309 (16 July 1985).
  7. I. Lundqvist, T. Pettersson, and G. Fladda, “Förfarande för att indikera fraktionsfördelningen hos suspenderade ämnen i ett strömmande medium,” Swedish patent 77 063 20-4 (1 March 1979).
  8. I. Lundqvist, T. Pettersson, and G. Fladda, “Method and apparatus for indicating the size distribution of particles in a flowing medium,” U.S. patent 4,318,180 (2 March 1982).
  9. S. Rydefalk and J. Einarsson, “Anordning för att i en suspension med åtminstone två typer av suspenderade ämnen var för sig mäta halten av varje ämnestyp,” Swedish patent 84 007 84-8 (3 March 1986).
  10. S. Rydefalk and J. Einarsson, “Device for separately measuring particles in a suspension,” U.S. patent 4,689,988 (1 September 1987).
  11. S. Rydefalk, “Fluctuations in the regular transmittance of dispersions of straight circular cylinders with a diameter much larger than the wavelength of the radiation,” J. Opt. Soc. Am. A 15, 1689-1697 (1998).
    [CrossRef]
  12. S. Rydefalk, “Theory of fluctuations in the regular transmittance through a dispersion of large cylindrical particles: extension to higher concentrations,” J. Opt. Soc. Am. A 16, 2737-2745 (1999).
    [CrossRef]
  13. S. Rydefalk, “Assessment of the mean and the variance of the random regular transmittance through a dispersion of large cylinders using numerical simulations,” Appl. Opt. 47, 993-1001(2008).
    [CrossRef] [PubMed]
  14. J. Gregory, “Turbidity fluctuations in flowing suspensions,” J. Colloid Interface Sci. 105, 357-371 (1985).
    [CrossRef]
  15. J. Hill, T. Pettersson, and S. Rydefalk, “The STFI long-fibre content meter and its process control applications,” Sven. Papperstidn. 80, 579-586 (1977).
  16. G. Fladda, T. Pettersson, L. Eriksson, and G. Tidstam, “A new optical method for measuring suspended solids in pulp and paper effluents,” in Proceedings of the 4th International Federation of Automatic Control Conference--Instrumentation and Automation in the Paper, Rubber, Plastics and Polymerisation Industries (International Federation of Automatic Control, 1980), pp. 9-22.
  17. S. Rydefalk, T. Pettersson, E. Jung, and I. Lundqvist, “The STFI optical fibre classifier,” in Proceedings of the International Mechanical Pulping Conference (Comité Européen de Liaison Pour la Cellulose et le Papier--European Liaison Committee for Pulp and Paper, 1981), Session III, No. 4, p. 16.
  18. T. Lindström, S. Rydefalk, and L. Wågberg, “The development of an integrated retention control system,” in Swedish Association of Pulp and Paper Engineers 84--The World Pulp and Paper Week. New Available Techniques (Swedish Association of Pulp and Paper Engineers, 1984), pp. 492-496.
  19. C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  20. J. Bendat and A. Piersol, Random Data Analysis and Measurement Procedures, 2nd ed. (Wiley, 1986).

2008 (1)

1999 (1)

1998 (1)

1986 (1)

J. Bendat and A. Piersol, Random Data Analysis and Measurement Procedures, 2nd ed. (Wiley, 1986).

1985 (1)

J. Gregory, “Turbidity fluctuations in flowing suspensions,” J. Colloid Interface Sci. 105, 357-371 (1985).
[CrossRef]

1984 (1)

T. Lindström, S. Rydefalk, and L. Wågberg, “The development of an integrated retention control system,” in Swedish Association of Pulp and Paper Engineers 84--The World Pulp and Paper Week. New Available Techniques (Swedish Association of Pulp and Paper Engineers, 1984), pp. 492-496.

1983 (1)

C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

1981 (1)

S. Rydefalk, T. Pettersson, E. Jung, and I. Lundqvist, “The STFI optical fibre classifier,” in Proceedings of the International Mechanical Pulping Conference (Comité Européen de Liaison Pour la Cellulose et le Papier--European Liaison Committee for Pulp and Paper, 1981), Session III, No. 4, p. 16.

1980 (1)

G. Fladda, T. Pettersson, L. Eriksson, and G. Tidstam, “A new optical method for measuring suspended solids in pulp and paper effluents,” in Proceedings of the 4th International Federation of Automatic Control Conference--Instrumentation and Automation in the Paper, Rubber, Plastics and Polymerisation Industries (International Federation of Automatic Control, 1980), pp. 9-22.

1977 (1)

J. Hill, T. Pettersson, and S. Rydefalk, “The STFI long-fibre content meter and its process control applications,” Sven. Papperstidn. 80, 579-586 (1977).

1954 (1)

A. A. Robertson and S. G. Mason, “Flocculation in flowing pulp suspensions,” Pulp Pap. Mag. Can. 55, 263-269 (1954).

1950 (1)

C. E. Hubley, A. A. Robertson, and S. G. Mason, “Flocculation in suspensions of large particles,” Can. J. Res. 28, 770-787(1950).

Bendat, J.

J. Bendat and A. Piersol, Random Data Analysis and Measurement Procedures, 2nd ed. (Wiley, 1986).

Bohren, C.

C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Einarsson, J.

S. Rydefalk and J. Einarsson, “Device for separately measuring particles in a suspension,” U.S. patent 4,689,988 (1 September 1987).

S. Rydefalk and J. Einarsson, “Anordning för att i en suspension med åtminstone två typer av suspenderade ämnen var för sig mäta halten av varje ämnestyp,” Swedish patent 84 007 84-8 (3 March 1986).

Eriksson, L.

T. Pettersson, G. Fladda, and L. Eriksson, “Förfarande för koncentrationsbestämning,” Swedish patent 75 135 24-4 (29 September 1977).

T. Pettersson, G. Fladda, and L. Eriksson, “Method for determination of concentration,” U.S. patent 4,110,044 (29 August 1978).

G. Fladda, T. Pettersson, L. Eriksson, and G. Tidstam, “A new optical method for measuring suspended solids in pulp and paper effluents,” in Proceedings of the 4th International Federation of Automatic Control Conference--Instrumentation and Automation in the Paper, Rubber, Plastics and Polymerisation Industries (International Federation of Automatic Control, 1980), pp. 9-22.

Fladda, G.

T. Pettersson, G. Fladda, and L. Eriksson, “Förfarande för koncentrationsbestämning,” Swedish patent 75 135 24-4 (29 September 1977).

T. Pettersson, G. Fladda, and L. Eriksson, “Method for determination of concentration,” U.S. patent 4,110,044 (29 August 1978).

I. Lundqvist, T. Pettersson, and G. Fladda, “Method and apparatus for indicating the size distribution of particles in a flowing medium,” U.S. patent 4,318,180 (2 March 1982).

G. Fladda, T. Pettersson, L. Eriksson, and G. Tidstam, “A new optical method for measuring suspended solids in pulp and paper effluents,” in Proceedings of the 4th International Federation of Automatic Control Conference--Instrumentation and Automation in the Paper, Rubber, Plastics and Polymerisation Industries (International Federation of Automatic Control, 1980), pp. 9-22.

I. Lundqvist, T. Pettersson, and G. Fladda, “Förfarande för att indikera fraktionsfördelningen hos suspenderade ämnen i ett strömmande medium,” Swedish patent 77 063 20-4 (1 March 1979).

Gregory, J.

J. Gregory, “Turbidity fluctuations in flowing suspensions,” J. Colloid Interface Sci. 105, 357-371 (1985).
[CrossRef]

Hill, J.

J. Hill, T. Pettersson, and S. Rydefalk, “The STFI long-fibre content meter and its process control applications,” Sven. Papperstidn. 80, 579-586 (1977).

Hubley, C. E.

C. E. Hubley, A. A. Robertson, and S. G. Mason, “Flocculation in suspensions of large particles,” Can. J. Res. 28, 770-787(1950).

Huffman, D.

C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Jung, E.

S. Rydefalk, T. Pettersson, E. Jung, and I. Lundqvist, “The STFI optical fibre classifier,” in Proceedings of the International Mechanical Pulping Conference (Comité Européen de Liaison Pour la Cellulose et le Papier--European Liaison Committee for Pulp and Paper, 1981), Session III, No. 4, p. 16.

Karlsson, H.

T. Pettersson and H. Karlsson, “Förfarande för att bestämma medelpartikelradie och/eller medelpartikellängd,” Swedish patent 81 058 02-6 (21 April 1988).

T. Pettersson and H. Karlsson, “Method for determining the average radius and/or the average length of particles carried by a flowing medium,” U.S. patent 4,529,309 (16 July 1985).

Lindstr, T.

T. Lindström, S. Rydefalk, and L. Wågberg, “The development of an integrated retention control system,” in Swedish Association of Pulp and Paper Engineers 84--The World Pulp and Paper Week. New Available Techniques (Swedish Association of Pulp and Paper Engineers, 1984), pp. 492-496.

Lundqvist, I.

I. Lundqvist, T. Pettersson, and G. Fladda, “Method and apparatus for indicating the size distribution of particles in a flowing medium,” U.S. patent 4,318,180 (2 March 1982).

S. Rydefalk, T. Pettersson, E. Jung, and I. Lundqvist, “The STFI optical fibre classifier,” in Proceedings of the International Mechanical Pulping Conference (Comité Européen de Liaison Pour la Cellulose et le Papier--European Liaison Committee for Pulp and Paper, 1981), Session III, No. 4, p. 16.

I. Lundqvist, T. Pettersson, and G. Fladda, “Förfarande för att indikera fraktionsfördelningen hos suspenderade ämnen i ett strömmande medium,” Swedish patent 77 063 20-4 (1 March 1979).

Mason, S. G.

A. A. Robertson and S. G. Mason, “Flocculation in flowing pulp suspensions,” Pulp Pap. Mag. Can. 55, 263-269 (1954).

C. E. Hubley, A. A. Robertson, and S. G. Mason, “Flocculation in suspensions of large particles,” Can. J. Res. 28, 770-787(1950).

Pettersson, T.

T. Pettersson, G. Fladda, and L. Eriksson, “Förfarande för koncentrationsbestämning,” Swedish patent 75 135 24-4 (29 September 1977).

T. Pettersson, G. Fladda, and L. Eriksson, “Method for determination of concentration,” U.S. patent 4,110,044 (29 August 1978).

T. Pettersson and H. Karlsson, “Förfarande för att bestämma medelpartikelradie och/eller medelpartikellängd,” Swedish patent 81 058 02-6 (21 April 1988).

I. Lundqvist, T. Pettersson, and G. Fladda, “Method and apparatus for indicating the size distribution of particles in a flowing medium,” U.S. patent 4,318,180 (2 March 1982).

S. Rydefalk, T. Pettersson, E. Jung, and I. Lundqvist, “The STFI optical fibre classifier,” in Proceedings of the International Mechanical Pulping Conference (Comité Européen de Liaison Pour la Cellulose et le Papier--European Liaison Committee for Pulp and Paper, 1981), Session III, No. 4, p. 16.

G. Fladda, T. Pettersson, L. Eriksson, and G. Tidstam, “A new optical method for measuring suspended solids in pulp and paper effluents,” in Proceedings of the 4th International Federation of Automatic Control Conference--Instrumentation and Automation in the Paper, Rubber, Plastics and Polymerisation Industries (International Federation of Automatic Control, 1980), pp. 9-22.

J. Hill, T. Pettersson, and S. Rydefalk, “The STFI long-fibre content meter and its process control applications,” Sven. Papperstidn. 80, 579-586 (1977).

T. Pettersson and H. Karlsson, “Method for determining the average radius and/or the average length of particles carried by a flowing medium,” U.S. patent 4,529,309 (16 July 1985).

I. Lundqvist, T. Pettersson, and G. Fladda, “Förfarande för att indikera fraktionsfördelningen hos suspenderade ämnen i ett strömmande medium,” Swedish patent 77 063 20-4 (1 March 1979).

Piersol, A.

J. Bendat and A. Piersol, Random Data Analysis and Measurement Procedures, 2nd ed. (Wiley, 1986).

Robertson, A. A.

A. A. Robertson and S. G. Mason, “Flocculation in flowing pulp suspensions,” Pulp Pap. Mag. Can. 55, 263-269 (1954).

C. E. Hubley, A. A. Robertson, and S. G. Mason, “Flocculation in suspensions of large particles,” Can. J. Res. 28, 770-787(1950).

Rydefalk, S.

S. Rydefalk, “Assessment of the mean and the variance of the random regular transmittance through a dispersion of large cylinders using numerical simulations,” Appl. Opt. 47, 993-1001(2008).
[CrossRef] [PubMed]

S. Rydefalk, “Theory of fluctuations in the regular transmittance through a dispersion of large cylindrical particles: extension to higher concentrations,” J. Opt. Soc. Am. A 16, 2737-2745 (1999).
[CrossRef]

S. Rydefalk, “Fluctuations in the regular transmittance of dispersions of straight circular cylinders with a diameter much larger than the wavelength of the radiation,” J. Opt. Soc. Am. A 15, 1689-1697 (1998).
[CrossRef]

T. Lindström, S. Rydefalk, and L. Wågberg, “The development of an integrated retention control system,” in Swedish Association of Pulp and Paper Engineers 84--The World Pulp and Paper Week. New Available Techniques (Swedish Association of Pulp and Paper Engineers, 1984), pp. 492-496.

S. Rydefalk, T. Pettersson, E. Jung, and I. Lundqvist, “The STFI optical fibre classifier,” in Proceedings of the International Mechanical Pulping Conference (Comité Européen de Liaison Pour la Cellulose et le Papier--European Liaison Committee for Pulp and Paper, 1981), Session III, No. 4, p. 16.

J. Hill, T. Pettersson, and S. Rydefalk, “The STFI long-fibre content meter and its process control applications,” Sven. Papperstidn. 80, 579-586 (1977).

S. Rydefalk and J. Einarsson, “Device for separately measuring particles in a suspension,” U.S. patent 4,689,988 (1 September 1987).

S. Rydefalk and J. Einarsson, “Anordning för att i en suspension med åtminstone två typer av suspenderade ämnen var för sig mäta halten av varje ämnestyp,” Swedish patent 84 007 84-8 (3 March 1986).

Tidstam, G.

G. Fladda, T. Pettersson, L. Eriksson, and G. Tidstam, “A new optical method for measuring suspended solids in pulp and paper effluents,” in Proceedings of the 4th International Federation of Automatic Control Conference--Instrumentation and Automation in the Paper, Rubber, Plastics and Polymerisation Industries (International Federation of Automatic Control, 1980), pp. 9-22.

Wågberg, L.

T. Lindström, S. Rydefalk, and L. Wågberg, “The development of an integrated retention control system,” in Swedish Association of Pulp and Paper Engineers 84--The World Pulp and Paper Week. New Available Techniques (Swedish Association of Pulp and Paper Engineers, 1984), pp. 492-496.

Appl. Opt. (1)

Can. J. Res. (1)

C. E. Hubley, A. A. Robertson, and S. G. Mason, “Flocculation in suspensions of large particles,” Can. J. Res. 28, 770-787(1950).

J. Colloid Interface Sci. (1)

J. Gregory, “Turbidity fluctuations in flowing suspensions,” J. Colloid Interface Sci. 105, 357-371 (1985).
[CrossRef]

J. Opt. Soc. Am. A (2)

Pulp Pap. Mag. Can. (1)

A. A. Robertson and S. G. Mason, “Flocculation in flowing pulp suspensions,” Pulp Pap. Mag. Can. 55, 263-269 (1954).

Sven. Papperstidn. (1)

J. Hill, T. Pettersson, and S. Rydefalk, “The STFI long-fibre content meter and its process control applications,” Sven. Papperstidn. 80, 579-586 (1977).

Other (13)

G. Fladda, T. Pettersson, L. Eriksson, and G. Tidstam, “A new optical method for measuring suspended solids in pulp and paper effluents,” in Proceedings of the 4th International Federation of Automatic Control Conference--Instrumentation and Automation in the Paper, Rubber, Plastics and Polymerisation Industries (International Federation of Automatic Control, 1980), pp. 9-22.

S. Rydefalk, T. Pettersson, E. Jung, and I. Lundqvist, “The STFI optical fibre classifier,” in Proceedings of the International Mechanical Pulping Conference (Comité Européen de Liaison Pour la Cellulose et le Papier--European Liaison Committee for Pulp and Paper, 1981), Session III, No. 4, p. 16.

T. Lindström, S. Rydefalk, and L. Wågberg, “The development of an integrated retention control system,” in Swedish Association of Pulp and Paper Engineers 84--The World Pulp and Paper Week. New Available Techniques (Swedish Association of Pulp and Paper Engineers, 1984), pp. 492-496.

C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

J. Bendat and A. Piersol, Random Data Analysis and Measurement Procedures, 2nd ed. (Wiley, 1986).

T. Pettersson, G. Fladda, and L. Eriksson, “Förfarande för koncentrationsbestämning,” Swedish patent 75 135 24-4 (29 September 1977).

T. Pettersson, G. Fladda, and L. Eriksson, “Method for determination of concentration,” U.S. patent 4,110,044 (29 August 1978).

T. Pettersson and H. Karlsson, “Förfarande för att bestämma medelpartikelradie och/eller medelpartikellängd,” Swedish patent 81 058 02-6 (21 April 1988).

T. Pettersson and H. Karlsson, “Method for determining the average radius and/or the average length of particles carried by a flowing medium,” U.S. patent 4,529,309 (16 July 1985).

I. Lundqvist, T. Pettersson, and G. Fladda, “Förfarande för att indikera fraktionsfördelningen hos suspenderade ämnen i ett strömmande medium,” Swedish patent 77 063 20-4 (1 March 1979).

I. Lundqvist, T. Pettersson, and G. Fladda, “Method and apparatus for indicating the size distribution of particles in a flowing medium,” U.S. patent 4,318,180 (2 March 1982).

S. Rydefalk and J. Einarsson, “Anordning för att i en suspension med åtminstone två typer av suspenderade ämnen var för sig mäta halten av varje ämnestyp,” Swedish patent 84 007 84-8 (3 March 1986).

S. Rydefalk and J. Einarsson, “Device for separately measuring particles in a suspension,” U.S. patent 4,689,988 (1 September 1987).

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Figures (11)

Fig. 1
Fig. 1

Experimental setup for measurement of the regular transmittance of optical radiation through a dispersion of particles.

Fig. 2
Fig. 2

Fitted model (solid curve) and simple model (dashed curve) of the response for φ as a function of particle length. The fitted model from [13] is based on numerical simulations. The simple model is an extrapolation of the special cases in Eq. (4). The response is expressed in k B L S a units, where k B L is k B for l l b .

Fig. 3
Fig. 3

Concentration meter characteristics. Response functions for k 1 φ S 1 (solid line) and k 1 φ F 1 (dashed curve) plotted together with a weighted PDF of a sample of wood pulp fibers (dotted curve, y axis to the right). The response is independent of particle length for almost all fiber lengths represented in this type of sample.

Fig. 4
Fig. 4

Concentration experiment 1. Concentration measurements on samples from group 1 (pine) and group 2 (birch). (a)  φ 1 and (b) A plotted against manually determined concentration.

Fig. 5
Fig. 5

Concentration experiment 2. (a)  φ 1 and (b) A plotted against manually determined concentration. The samples from groups 3 to 8 are shown.

Fig. 6
Fig. 6

Discrimination of small particles in the k 1 φ 1 signal is illustrated using papermaking pulp as large particles and filler clay as small particles. At two pulp concentrations, clay was added without changing the pulp concentration. k 1 φ 1 is plotted against manually determined concentration. The small clay particles add to the total concentration but have almost no effect on the k 1 φ 1 signal.

Fig. 7
Fig. 7

Length classifier. Length classifier responses for S short , S medium , and S long in Eqs. (21, 22, 23) according to both the simple (dashed curve) and the fitted (solid curve) models plotted together with the same PDF as in Fig. 3 (dotted curve, y axis to the right) in order to show how much of the total concentration may be found in each of the length classes.

Fig. 8
Fig. 8

Length measurement. Length response function r ( l ) for m L = l 3 k 3 φ 3 / ( k 1 φ 1 ) according to both the simple (dashed curve) and the fitted (solid curve) models plotted together with the same PDF as in Fig. 3 (dotted curve, y axis to the right) showing that the mean length is underestimated by the simple model since the longest particles of the population exceed l 3 .

Fig. 9
Fig. 9

Comparison of three length measurement methods: (1) microscope method, (2) Kajaani-FS200, and (3) the length signal φ 3 / φ 1 from the triple-beam unit. (1), (2), and (3) are plotted versus each other in three combinations in (a), (b), and (c).

Fig. 10
Fig. 10

Particle width measurement. Response for different fiber widths (fiber diameters) from a particle width meter based on φ 1 / A according to both the simple (dashed curve) and the fitted (solid curve) models plotted together with the PDF (dotted curve, y axis to the right) for a width distribution calculated from the length distribution in Fig. 3 using a simplified assumption regarding the width–length dependence. A proper width signal was obtained for almost all particles of the population.

Fig. 11
Fig. 11

Width signal from the triple beam unit. (a) Comparison with microscopically determined particle widths for samples where the fines fraction has been removed. (b) Comparison with the fines fraction contents of four otherwise identical samples.

Tables (3)

Tables Icon

Table 1 Descriptions of Sample Groups a

Tables Icon

Table 2 Length Classifier Output from the Numerical Experiment

Tables Icon

Table 3 Moments of Λ for Two Extreme Orientation States

Equations (25)

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μ T = 1 k A p ,
σ T 2 = k B p ,
k A = 4 Q μ Λ / ( π ρ p d ) ,
k B = { 16 Q 2 l E ( Λ 2 ) / ( π 2 ρ p D b 2 ) l l b 128 Q 2 μ Λ / ( 3 π 3 ρ p D b ) l l b ,
l b = 8 μ Λ D b 3 π E ( Λ 2 ) .
μ T = exp ( k A p ) ,
σ T 2 = exp ( 2 k A p ) [ exp ( k B p ) 1 ] ,
V 2 = σ T 2 μ T 2 = exp ( k B p ) 1.
φ = In ( V 2 + 1 ) ,
φ = k B p .
φ = ν ln ( V 2 ν + 1 )
A = ln μ T = k A p
R i = φ i , I A i , I / φ i , II A i , II ,
R ¯ = 1 q i = 1 q R i
T k = 1 C b ( C 1 + C 2 + + C N ) ,
T k = G k C b ( Q k , 1 + Q k , 2 + + Q k , N ) .
μ T k = 1 3 μ Q k a S 2 ρ p , k d k ,
σ T k 2 = 3 E ( Q k 2 ) a d k S 2 ρ p , k D b 2 .
variance   ratio = σ T k 2 σ T 2 = 9 π 256 E ( Q k 2 ) Q 2 ρ p ρ p , k d k D b
attenuation   ratio = 1 μ T k 1 μ T = 3 π 8 μ Q k Q ρ p ρ p , k d d k ,
S short = l 2 l 2 l 1 ( k 1 φ 1 k 2 φ 2 ) ,
S medium = l 1 l 1 l 2 k 1 φ 1 + l 2 ( l 1 l 3 ) l 2 2 l 2 l 3 l 1 l 2 + l 1 l 3 k 2 φ 2 l 3 l 3 l 2 k 3 φ 3 ,
S long = - l 2 l 3 l 2 k 2 φ 2 + l 3 l 3 l 2 k 3 φ 3 ,
m L = 0 r ( l ) f L ( l ) d l ,
1 μ 1 D = 3 π 2 D b 32 Q φ 1 A ,

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